mordkovitch-gdz-7-2003_1-1145 (542425), страница 12
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Ответ: ; − .55554 2 164 ⎞⎛ 24⎞24 24⎛2b −= 0; ⎜ b − ⎟⎜ b + ⎟ = 0; b = ; b = −б)11 ⎠⎝ 711 ⎠49121711 711⎝73333b = 1 ; b = – 1 . Ответ: 1 ; – 1 .111111119 ⎞⎛ 39⎞9 2 8139 39⎛3= 0; ⎜ c − ⎟⎜ c + ⎟ = 0; c = ; c = − ;в)c −10 ⎠⎝ 410 ⎠16100410 410⎝41111c = 1 ; c = – 1 . Ответ: 1 ; – 1 .55558 ⎞⎛ 68⎞36 264⎛ 6d –= 0; ⎜ d − ⎟⎜ d + ⎟ = 0;г)21 ⎠⎝ 3521 ⎠1225441⎝ 3568 682222; d = − ; d = 2 ; d = – 2 . Ответ: 2 ; – 2 .d=3521 35219999a=№ 624а) (2x – 5)2 – 36 =0;(2x – 41)(2x + 31) = 0;2x = 41; 2x = – 31;11x = 20 ; x = – 15 .2211Ответ: 20 ; – 15 .22в) (5z – 3)2 – 9z2 = 0;(2z – 3)(14z – 3) = 0;2z = 3; 14z = 3;13z=1 ;z=.2143.Ответ: 1;14б) (4 – 11y)2 – 1= 0;(3 – 11y)(5 – 11y) = 0;11y = 3; 11y = 5;35y=;y=.11113 5Ответ:; .11 112г) (4t – 3) – 25t2 = 0;(– t – 3)(9t – 3) = 0;– t = 3; 9t = 3;1t = – 3; t = .31Ответ: – 3; .3101№ 625а) (a + 1)2 – (2a + 3)2 = 0;(– a – 2)(3a + 4) = 0;a = – 2;3a = – 4;1a=–1 .31.3в) (5c + 8)2 – (c – 10)2 = 0;(4c + 18)(6c – 2) = 0;4c = – 18;6c = 2;1c=–4 ;2Ответ: – 2; – 1c = 3.Ответ: – 4а)б)в)г)б) (3b – 2)2 – (b + 1)2 = 0;(2b – 3)(4b – 1) = 0;2b = 3;4b = 1;1b=1 ;21b= .41 1Ответ: 1 ; .2 4г) (7d – 13)2 – (9d – 25)2 = 0;(– 2d + 12)(16d – 38) = 0;2d = 12;16d = 38;d = 6;d =21; 3.23.8Ответ: 6; 23.8№ 6261 3 8 3 ⎛12 ⎞⎛ 114 ⎞a − b = ⎜ a − b ⎟⎜ a 2 + ab + b 2 ⎟ ;8273 ⎠⎝ 439 ⎠⎝264 3 729 3 ⎛ 89 ⎞⎛ 643681 2 ⎞c +d = ⎜ c + d ⎟⎜ c 2 − cd +d ⎟;34310007104935100⎝⎠⎝⎠125 3 216 3 ⎛ 56 ⎞⎛ 25 2 1536 2 ⎞x −y = ⎜ x − y ⎟⎜ x +xy +y ⎟;5123437 ⎠⎝ 642849 ⎠⎝81 3 125 3 ⎛ 15 ⎞⎛ 1525 ⎞m +n = ⎜ m + n ⎟⎜ m 2 − mn + n 2 ⎟ .7292166 ⎠⎝ 815436 ⎠⎝9№ 627а) a6 – 8 = (a2)3 – 23 = (a2 – 2)(a4 + 2a2 + 4);б) 27 + b9 = 33 + (b3)3 = (3 + b3)(9 – 3b3 + b6);2( )3 ⎛11⎛1⎞⎞⎛ 1 1⎞− x 6 = ⎜ ⎟ − x 2 = ⎜ − x 2 ⎟⎜ + x 2 + x 4 ⎟ ;8⎝2⎠⎝2⎠⎝ 4 2⎠1⎛ 1⎞ ⎛ 1⎞⎛ 1 1⎞– x6 = − ⎜ + y 6 ⎟ = ⎜ − − y 2 ⎟⎜ − y 2 + y 4 ⎟ .г) –64⎝ 64⎠ ⎝ 4⎠⎝ 16 4⎠в)102№ 628а) x3y3 – c3 = (xy)3 – c3 = (xy – c)(x2y2 + xyc + c2);б) a3 + m3n9 = a3 + (mn3)3 = (a + mn3)(a2 – amn3 + m2n6);в) m6n3 – p12 = (m2n)3 – (p4)3 = (m2n – p4)(m4n2 + m2np4 + p8);г) q3 + c15d18 = q3 + (c5d6)3 = (q + c5d6)(q2 – qc5d6 + c10d12).№ 6293( )а)1 6⎛1 ⎞a − b9 = ⎜ a 2 ⎟ − b38⎝2 ⎠1⎛1⎞⎛ 1⎞= ⎜ a 2 − b3 ⎟⎜ a 4 + a 2b3 + b6 ⎟ ;2⎝2⎠⎝ 4⎠б)8 3a + x9 =27в)1 3⎛1 ⎞x + y6 = ⎜ x ⎟ + y2125⎝5 ⎠г)64 3 343 6 ⎛ 4 ⎞ ⎛ 7 ⎞7 ⎞⎛ 161449 4 ⎞⎛4m −n = ⎜ m ⎟ − ⎜ n ⎟ = ⎜ m − n2 ⎟⎜ m2 + mn2 +n ⎟.729100010 ⎠⎝ 8145100 ⎠⎝ 9 ⎠ ⎝ 10 ⎠⎝933⎛2 ⎞⎛23 33 ⎞⎛ 4 2 2 36⎞⎜ a ⎟ + (x ) = ⎜ a + x ⎟⎜ a − ax + x ⎟ ;3⎝3 ⎠⎝3⎠⎝ 9⎠3( )331⎛1⎞⎛ 1⎞= ⎜ x + y 2 ⎟⎜ x 2 − xy 2 + y 4 ⎟ ;5⎝5⎠⎝ 25⎠6№ 630а)(2с + 1)3 – 64 = (2c + 1)3 – 43 =(2c + 1 – 4)((2c + 1)2 + 4(2c + 1)+42) == (2c – 3)(4c2 + 4c + 1 + 8c + 4 + 16) = (2c – 3)(4c2 + 12c + 21);б) (3p – 4)3 + 1 = (3p – 4 + 1)((3p – 4)2 – 3p + 4 + 1) == (3p – 3)(9p2 – 24p + 16 – 3p + 4 + 1) = 3(p – 1)(9p2 – 27p + 21) == 9(p – 1)(3p2 – 9p + 7);в) 8 – (3 – k)3 = (k – 1)(4 + 6 – 2k + 9 – 6k + k2) = (k – 1)(k2 – 8k + 19);г) (5a + 4)3 – 27 = (5a + 1)(25a2 + 40a + 16 + 15a + 12 + 9) == (5a + 1)(25a2 + 55a + 37).№ 631а) (6b + 8)3 – 125 = (6b + 8 – 5)((6b + 8)2 + 5(6b + 8) + 25) == 9(2b + 1)(12b2 + 42b + 43);б) 1000 + (3q + 12)3 = (3q + 12)(100 – 10(3q + 12) + (3q + 12)2) == (3q + 12)(100 – 30q – 120 + 9q2 + 72q + 144) =3(q + 4)(9q2 + 42q + 124);в) 8x3–(5x–3)3=(2x–5x+3)((2x)2 + 2x(5x – 3) + (5x–3)2)=– 3(x–1)(4x2+10x2 –– 6x + 25x2 – 30x + 9) = – 9(x – 1)(13x2 – 12x + 3);г)(3x + 2y)3 + 729y3 = (3x + 2y + 9y)((3x + 2y)2 – 9y(3x + 2y) + 81y2) == (3x+11y)(9x2+12xy+4y3–27xy–18y2+81y2)= (3x + 11y)(9x2 – 15xy + 67y2).№ 6322229 2164 ⎞⎛3 ⎞⎛4 ⎞⎛3a − 2ab + b 2 = ⎜ a ⎟ − 2ab + ⎜ b ⎟ = ⎜ a − b ⎟ ;1693 ⎠⎝4 ⎠⎝3 ⎠⎝4222⎞⎛3 2 5 2⎞⎛9 6 2 4 4 25 2 6 2 2 ⎛ 3 2 ⎞2 2⎜ a + b ⎟2 2 ⎛5 2⎞⎜⎟б)=aba b +a b + a b = a b ⎜ a ⎟ +a b +⎜ b ⎟6 ⎠ ;⎝5⎜⎝ 5 ⎠2536⎝ 6 ⎠ ⎟⎠⎝а)103221 41 ⎞⎛1 ⎞⎛a =(b4)2 + a2b4 + ⎜ a 2 ⎟ = ⎜ b 4 + a 2 ⎟ ;2 ⎠4⎝2 ⎠⎝4222 222г) 0,01x + y – 0,2x y = (0,1x ) – 0,2x y + y = (0,1x2 – y)2.в) b8 + a2b4 +№ 633а) 513 – 263 = (51 – 26)(512 + 51 · 26 + 262) = 25 · (512 + 51 · 26 + 262) 25 –это сомножитель, значит, выражение делится на 25;б) 433 + 173 = (43 + 17)(432 – 43 · 17 + 172) = 60(432 – 43 · 17 + 172) 60 –это сомножитель, значит, выражение делится на 60;в) 543 – 143 = (54 – 14)(542 – 54 · 14 + 142) = 40(542 – 54 · 14 + 142) 40 –это сомножитель, значит, выражение делится на 40;г) 383 + 373 = (38 + 37)(382 – 38 · 37 + 372) = 75(382 – 38 · 37 + 372) 75 –это сомножитель, значит, выражение делится на 75.№ 634а) (532+222–472–162) : (652 – 2 · 65 · 59 + 592) = (532 – 472 + 222 – 162) :: (65 – 59)2 = ((53 – 47)(53 + 47) + (22 – 16)(22 + 16)) : 62 == (100 · 6 + 38 · 6) : 36 = 6 · (100 + 38) : 36 = 138 : 6 = 23;б)593 − 413+ 59 · 41 = (59 – 41)(592 + 59 · 41 + 412) : 18 + 59 · 41 =18592 + 2 · 59 · 41 + 412 = (59 + 41)2 = 1002 = 10000;в) (1092 – 2 · 109 · 61 + 612) : (792 + 732 – 492 – 552) == (109 – 61)2 : ((79 – 49)(79 + 49) + (73 – 55)(73 + 55)) == 482 : (30 · 128 + 18 · 128) = 482 : (128· (30 + 18)) = 48 : 128 =г)3;8673 + 523– 67 · 52 = (67 + 52)(672 – 67 · 52 + 522) : 119 – 67 · 52 =119= 672 – 2 · 67 · 52 + 522 = (67 – 52)2 = 152 = 225.№ 635(⎛ ( 97 − 53) 97 2 + 97 ⋅ 53 + 532⎛ 973 − 533⎞22+ 97 ⋅ 53 ⎟ : (152,5 – 27,5 ) = = ⎜а) ⎜⎜⎟⎜⎜4444⎝⎠⎝: ((152,5–27,5)(152,5+27,5)) = (972 +2 · 97 · 53 + 532) : (125 · 180) == (97 + 53)2 : 22500 = 1502 : 22500 = 1;⎛ 573 + 333б) (36,52 – 27,52) : ⎜⎜⎝90⎞− 57 ⋅ 33 ⎟ =⎟⎠(⎛ ( 57 + 33) 57 2 − 57 ⋅ 33 + 332= ((36,5–27,5)(36,5+27,5)) ⎜⎜⎜⎝90) − 57 ⋅ 33 ⎞⎟ == (9 · 64) : (572 – 2 · 57 · 33 + 332) = 576 : (57 – 33)2 = 1;104⎟⎟⎠) ⎞⎟ :⎟⎟⎠()⎛ ( 79 − 41) 792 + 79 ⋅ 41 + 412⎞⎛ 793 − 413⎞+ 79 ⋅ 41⎟ : (133,52 – 58,52) = ⎜в) ⎜+ 79 ⋅ 41⎟⎜ 38⎟⎜⎜⎟⎟38⎝⎠⎝⎠· (133,5–58,5)(133,5+ 58,5) = (79 + 41)2 : (75 · 192) = 1202 : 14400 = 1;⎛ 693 + 293г) (94,52 – 30,52) : ⎜⎜⎝98⎞− 69 ⋅ 29 ⎟ =⎟⎠⎛ (69 + 29)(692 − 69 · 29 + 292 )⎞69 · 29 ⎟ =⎟98⎝⎠= (94,5 – 30,5)(94,5 + 30,5) ⎜⎜= 64 · 125 : (69 – 29)2 = 8000 : 1600 = 5.№ 636а) a2 + ∗ + b2 = (a + b)2;б) b2 + 20ab + ∗ = (b + 10)2;в) ∗ – 56a + 49 = (4a – 7)2;г) ∗1 – 12c + ∗2 = (3c – 2)2;(a + b)2 = a2 + 2ab + b2; ∗ = 2ab;(b + 10)2 = b2 + 20ab + 100; ∗ = 100;(4a – 7)2 = 16a2 – 56a + 49; ∗ = 16a;(3c – 2)2 = 9c2 – 12c + 4; ∗1 = 9c2, ∗2 = 4.№ 637а) b2 – 20b + ∗1 = (∗2 – 10)2; (∗2–10)2=∗22 – 2 · ∗2 · 10 + 100; ∗1=100, ∗2=b;б) ∗1 – 42pq+49q2 = (3p – ∗2)2; (3p–∗2)2=9p2–2 · ∗2 · 3p+∗22; ∗1=9p2, ∗2=7q;11в) 25a2 + ∗1 + b2 = (∗2 + b)2;421 21 22(∗2 + b) = ∗2 + ∗2b + b ; ∗2 = 5a, ∗1 = ∗2 · b = 5ab;24г) 0,01b2 + ∗1 + 100c2 = (0,1b + ∗2)2; (0,1b + ∗2)2 = 0,01b2 – 0,2b · ∗2 + ∗22;∗2 = 10c, ∗1 = ∗2 · 0,2b = 10c · 0,2b = 2bc.№ 638а) ∗1 + 56ab + 49b2 = (4a + ∗2)2;(4a + ∗2)2 = 16a2 + 2 · ∗2 · 4a + ∗22; ∗1 = 16a2, ∗2 = 7b;б) 225x2 – ∗1 + 64y2 = (15x – ∗2)2;(15x – ∗2)2 = 225x2 – 2 · 15x · ∗2 + ∗22;∗2 = 8y, ∗1 = 30x · ∗2 = 30x · 8y = 240xy;в) ∗1 + 96xy + 36y2 = (8x + ∗2)2;(8x + ∗2)2 = 64x2 + 16x · ∗2 + ∗22; ∗1 = 64x2, ∗2 = 6y;г) 100a2 + ∗1 + 49b2 = (10a + ∗2)2;(10a + ∗2)2 = 100a2 + 20a · ∗2 + ∗22; ∗2 = 7b, ∗1 = 140ab.№ 639а) m2 + 40m + ∗1 = (∗2 + 20)2;(∗2 + 20)2 = ∗22 + 2 · 20 · ∗2 + 400; ∗1 = 400, ∗2 = m;б) ∗1 – 70pq + ∗2 = (7p – ∗3)2;(7p – ∗3)2 = 49p2 – 2 · 7p · ∗3 + ∗32;10570 pq= 5q, ∗2 = 25q2;14 pв) ∗1 + 42ac + 49c2 = (∗2 + ∗3)2;21ac∗3 = 7c, 2 · ∗2 · ∗3 = 42ac, ∗2 == 3a, ∗1 = ∗22 = 9a2;7cг) 25z2 – ∗1 + ∗2 = (∗3 – 8t)2;∗2 = (8t)2 = 64t2, ∗3 = 5z, ∗1 = 5z · 8t · 2 = 80zt.∗1 = 49p2, ∗3 =§ 23.
Разложение многочлена на множителис помощью комбинаций различных приемов№ 640а) 5x2 – 5 = 5(x2 – 1) = 5(x – 1)(x + 1);б) 10x2 – 10y2 = 10(x2 – y2) = 10(x – y)(x + y);в) 3a2 – 12 = 3(a2 – 4) = 3(a – 2)(a + 2);г) 9b3 – b = b(9b2 – 1) = b(3b – 1)(3b + 1).№ 641а) 9x2 – 81x = 9x(x – 9);б) y3 – 100y = y(y2 – 100) = y(y – 10)(y + 10);в) 64a – a3 = a(64 – a2) = a(8 – a)(8 + a);г) b3 – 144b = b(b2 – 122) = b(b – 12)(b + 12).№ 642а) c3 – 25c = c(c2 – 25) = c(c – 5)(c + 5);б) 50m – 2n2m = 2m(25 – n2) = 2m(5 – n)(5 + n);в) 0,04s – sa2 = s(0,04 – a2) = s(0,2 – a)(0,2 + a);г)16 2⎛ 16 2⎞⎛4⎞⎛ 4⎞p q − q3 = q ⎜p − q 2 ⎟ = q ⎜ p − q ⎟⎜ p + q ⎟ .49⎝ 49⎠⎝7⎠⎝ 7⎠№ 643а) 5a2 + 10ab + 5b2 = 5(a2 + 2ab + b2) = 5(a + b)2;б) 2x2 + 4xy + 2y2 = 2(x2 + 2xy + y2) = 2(x + y)2;в) 3m2 + 3n2 – 6mn = 3(m2 – 2mn + n2) = 3(m – n)2;г) 8n2 – 16n + 8 = 8(n2 – 2n + 1) = 8(n – 1)2.№ 644а) –3x2 + 12x – 12 = –3(x2 – 4x + 4) = –3(x – 2)2;б) –2a2 + 20ab – 50b2 = –2(a2 – 10ab + 25b2) = –2(a – 5b)2;в) –5p2 – 10pq – 5q2 = –5(p2 + 2pq + q2) = –5(p + q)2;г) –12z3 – 12z2 – 3z = –3z(4z2 + 4z + 1) = –3z(2z + 1)2.№ 645а) a4 – 16 = (a2)2 – 42 = (a2 – 4)(a2 + 4) = (a – 2)(a + 2)(a2 + 4);б) b4 – 81 = (b2 – 9)(b2 + 9) = (b – 3)(b + 3)(b2 + 81);106в) y8–1=(y4 – 1)(y4 + 1) = (y2 – 1)(y2 + 1)(y4 + 1)=(y–1)(y+1)(y2 + 1)(y4 + 1);г) x4 – z4 = (x2 – z2)(x2 + z2) = (x – z)(x + z)(x2 + z2).№ 646а) 4m3 – 4n3 = 4(m3 – n3) = 4(m – n)(m2 + mn + n2);б) 13a3 + 13b3 = 13(a3 + b3) = 13(a + b)(a2 – ab + b2);в) 15c3 + 15d3 = 15(c3 + d3) = 15(c + d)(c2 – cd + d2);г) 21s3 – 21t3 = 21(s3 – t3) = 21(s – t)(s2 + st + t2).№ 647а) 6x5y – 24xy3 = 6xy(x4 – 4y2) = 6xy(x2 – 2y)(x2 + 2y);б) 3a4b2 + 24ab5 = 3ab2(a3 + 8b3) = 3ab(a + 2b)(a2 – 2ab + 4b2;в) 0,3y2 – 2,7y6 = 0,3y2(1 – 9y4) = 0,3y2(1 – 3y2)(1 + 3y2);г) 0,1x4y – 2,7xy4 = 0,1xy(x3 – 27y3) = 0,1xy(x – 3y)(x2 + 3xy + 9y2).№ 648а) (m + 3)3 – 8 = (m + 3 – 2)((m + 3)2 + 2(m + 3) + 4) == (m + 1)(m2 + 6m + 9 + 2m + 6 + 4) = (m + 1)(m2 + 8m + 19);б) (c–1)3+27 = (c – 1 + 3)(c2 – 2c + 1 – 3c + 3 + 9) = (c + 2)(c2 – 5c + 12);в) (a–12)3–125 = (a–12–5)(a2–24a+144+5a–60+25)=(a – 17)(a2–19a+109);г) (b+4)3+64=(b+4+4)(b2+8b+16+4b+16+16) = (b + 8)(b2 + 12b + 48).№ 649а) (x2 + 1)2 – 4x2 = (x2 + 1 – 2x)(x2 + 1 + 2x) = (x – 1)2(x + 1)2;б) (y2 + 2y)2 – 1 = (y2 + 2y – 1)(y2 + 2t + 1) = (y2 + 2y – 1)(y + 1)2;в) 81 – (c2 + 6c)2 = (9 – c2 – 6c)(9 + c2 + 6c) = (9 – c2 – 6c)(c + 3)2;г) 16m2 – (m – n)2 = (4m – m + n)(4m + m – n) = (3m + n)(5m – n).№ 650а) (a2 + 2ab + b2) – c2 = (a + b)2 – c2 = (a + b – c)(a + b +c);б) 16 – (x2 – 2xy + y2) = 16 – (x – y)2 = (4 – x + y)(4 + x – y);в) 1 – m2 – 2mn – n2 = 1 – (m + n)2 = (1 – m – n)( 1 + m + n);г) 4 – p2 – 2pq – q2 = 4 – (p + q)2 = (2 – p – q)(2 + p + q).№ 651а) x2 – 2xc + c2 – d2 = (x – c)2 – d2 = (x – c – d)(x – c + d);б) a2 + 2a – b2 + 1 = (a + 1)2 – b2 = (a + 1 – b)(a + 1 + b);в) c2 – d2 + 6c + 9 = (c + 3)2 – d2 = (c + 3 – d)(c + 3 + d);г) r2 – s2 – 10s – 25 = r2 – (s + 5)2 = (r – s – 5)(r + s + 5).№ 652а) x2 + 2xy – m2 + y2 = (x + y)2 – m2 = (x = y – m)(x = y + m);б) c2 – a2 + 2ab – b2 = c2 – (a – b)2 = (c – a + b)(c + a – b);в) m2 – n2 – 8m + 16 = (m – 4)2 – n2 = (m – 4 – n)(m – 4 + n);г) 9 – p2 + q2 – 6q = (q – 3)2 – p2 = (q – 3 – p)(q – 3 + p).107№ 653а) x3 – x2y – xy2 + y3 = x3 + y3 – x2y – xy2 = (x + y)(x2 – xy + y2) – xy(x + y) == (x + y)(x2 – xy + y2 – xy) = (x + y)(x2 – 2xy + y2) = (x + y)(x – y)2;б) a3 + a2b – ab2 – b3 = (a – b)(a2 + ab + b2) + ab(a – b) == (a – b)(a2 + 2ab + b2) = (a – b)(a + b)2;в) c2+2c–d2+2d=c2–d2+2 (c + d) = (c – d)(c + d) + 2 (c + d)=(c+d)(c – d + 2);г) m2 – 2n – m – 4n2 = m2 – 4n2 – (2n + m) == (m – 2n)(m + 2n) – (2n + m) = (2n + m)(m – 2n – 1).№ 654а) x2(x – 3) – 2x(x – 3) + (x – 3) = (x – 3)(x2 – 2x + 1) = (x – 3)(x – 1)2;б) (1 – a)2 – 4a(1 – a)2 + 4a(1 – a)2 = (1 – a)2(1 – 4a + 4a) = (1 – a)2.№ 655а) a3 + 8b3 + a2 – 2ab + 4b2 = (a + 2b)(a2 – 2ab + 4b2) + a2 – 2ab + 4b2 == (a + 2b + 1)(a2 – 2ab + 4b2);б) 8c3 – d3 + 4c2 + 2cd + d2 = (2c – d)(4c2 + 2cd + d2) + 4c2 + 2cd + d2 == (2c – d + 1)(4c2 + 2cd + d2).№ 656а) x3 + 8y3 + x2 + 4xy + 4y2 = (x + 2y)(x2 – 2xy + 4y2 ) + (x + 2y)2 == (x + 2y)(x2 – 2xy + 4y2 + x + 2y);б) 8p3 – q3 + 4p2 – 4pq + q2 = (2p – q)(4p2 + 2pq + q2) + (2p – q)2 == (2p – q)(q2 + 2pq + 4p2 + 2p – q).№ 657а) a3 – a2 – 2a + 8 = a3 + 8 – a(a + 2) = (a + 2)(a2 – 2a + 4) – a(a + 2) == (a + 2)(a2 – 2a + 4 – a) = (a + 2)(a2 – 3a + 4);б) b3 – 6b2 – 6b + 1 = b3 + 1 – 6b(b + 1) = (b + 1)(b2 – b + 1) – 6b(b + 1) == (b + 1)(b2 – b + 1 – 6b) = (b + 1)(b2 – 7b + 1).№ 658а) x2 – 10x + 24 = x2 – 10x + 25 – 1 = (x – 5)2 – 1 = (x – 6)(x – 4);б) y2 – 14y + 40 = y2 – 14y + 49 – 9 = (y – 7)2 – 9 = (y – 10)(y – 4);в) b4 + 4b2 – 5 = b4 + 4b2 + 4 – 9 = (b2 + 2)2 – 9 == (b2 + 2 – 3)(b2 + 2 + 3) = (b2 – 1)(b2 + 5) = (b – 1)(b + 1)(b2 + 5);г) a2 – 6a + 5 = a2 – 6a + 9 – 4 = (a – 3)2 – 4 = (a – 5)(a – 1).№ 659а) 4a2 – 12ab + 5b2 = 4a2 – 12ab + 9b2 – 4b2 = (2a – 3b)2 – 4b2 == (2a – 5b)(2a – b);б) 9c2 – 24cd + 7d2 = 9c2 – 24cd + 16d2 – 9d2 = (3c – 4d)2 – 9d2 == (3c – 7d)(3c – d);в) 25a2 – 20ab – 12b2 = 25a2 – 20ab + 4b2 – 16b2 = (5a + 2b)2 – 16b2 == (5a – 2b)(5a + 6b);г) 9m2 – 30mk + 16k2 = 9m2 – 30mk + 25k2 – 9k2 = (3m – 5k2) – 9k2 == (3m – 8k)(3m – 2k).108№ 660а) a2 + 7a + 10 = a2 + 5a + 2a + 10 = a(a + 5) + 2(a + 5) = (a + 2)(a + 5);б) x4+7x2+12 = x4 + 3x2 + 4x2 + 12 = x2(x2 + 3) + 4(x2 + 3) = (x2 + 4)(x2 + 3);в) b2 – 3b – 4 = b2 – 1 – 3b – 3 = (b – 1)(b + 1) – 3(b + 1) = (b – 4)(b + 1) г)y4 – 5y2 + 4 = y4 – 4y2 – y2 + 4 = y2(y2 – 4) – y2 – 4 == (y2 – 1)(y2 – 4) = (y – 1)(y + 1)(y – 2)(y + 2).№ 661а) x2+5xy+6y2=x2 + 2xy + 3xy + 6y2 = x(x + 2y) + 3y(x + 2y)=(x+3y)(x + 2y);б) 4m2–5mn+n2=4m2–4mn – mn + n2 = 4m(m – n) + n(n – m)=(m–n)(4m – n);в) p2–pq–2q2=p2+pq–2q2 – 2pq = p(p + q) – 2q(p + q) = (p + q)(p – 2q);г) a2+7ab+6b2=a2 + ab + 6ab + 6b2 = a(a + b) + 6b(a + b) = (a + b)(a + 6b).№ 662а) x3 – x = 0;x(x2 – 1) = 0;x(x – 1)(x +1) = 0;x = 0, x = 1, x = –1.Ответ: 0; 1; –1.в) c3 + c2 = 0;c2(c + 1) = 0;c = 0, c = –1.Ответ: 0; –1.б) 16y – y3 = 0;y(16 – y2) = 0;y(4 – y)(4 + y) = 0;y = 0, y = 4, y = –4.Ответ: 0; 4; –4.г) d3 + d = 0;d(d2 + 1) = 0;d = 0, d2 + 1 ≠ 0 не при каких d.Ответ: 0.№ 663а) x3 + x2 – 4x – 4 = 0;x2(x + 1) – 4(x + 1) = 0;(x2 – 4)(x + 1) = 0;(x – 2)(x + 2)(x + 1) = 0;x = 2, x = –2, x = –1.Ответ: 2; –2; –1.в) 9z + 9 – z3 – z2 = 0;9(z + 1) – z2(z + 1) = 0;(9 – z2)(z + 1) = 0;(3 – z)(3 + z)(z + 1) = 0;z = 3, z = –3, z = –1.Ответ: 3; –3; –1.б) y3 + 2y2 – 4y – 8 = 0;y2(y + 2) – 4(y + 2);(y2 – 4)(y + 2) = 0;(y – 2)(y + 2)2 = 0;y = 2, y = – 2.Ответ: 2; –2.г) p3 – p2 – 4p + 4 = 0;p2(p – 1) – 4(p – 1) = 0;(p2 – 4)(p – 1) = 0;(p – 2)(p + 2)(p – 1) = 0;p = 2, p = –2, p = 1.Ответ: 2; –2; 1.№ 664x1 + x2 = 7; x1 · x2 = 2;а) x1x22 + x12x2 = x1x2(x1 + x2) = 2 · 7 = 14;б) (x1 = x2)2 = 72 = 49;в) x12 + x22 = x12 + x22 + 2x1x2 – 2x1x2 = (x1 + x2)2 – 2x1x2 = 49 – 4 = 45;г) (x13 + x23) = (x1 + x2)(x12 – x1x2 + x22) == (x1 + x2)((x1 + x2)2 – 3x1x2) = 7(49 – 6) = 7 · 43 = 301.109№ 665x1 + x2 = 5; x1 · x2 = –3а) x14 + x24 = x14 + 2x12x22 + x14 – 2x12x22 = (x12 + x22)2 – 2x12x22 == ((x1 + x2)2 – 2x1x2)2 – 2x12x22 = (25 + 6)2 – 18 = 312 – 18 = 961 – 18 = 943;б) (x1 – x2)2 = x12 – 2x1x2 + x22 = (x1 + x2)2 – 4x1x2 = 25 + 12 = 37;в) x13x22 + x12x23 = x12x22(x1 + x2) = 9 · 5 = 45;г) x12x24 + x14x22 = x12x22(x12 + x22) = x12x22(x12 + 2x1x2 + x22 – 2x1x2) == x12x22((x1 + x2)2 – 2x1x2) = 9 · (25 + 6) = 279.§ 24.
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